Related papers: Real multiplication revisited

Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert…

In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over…

Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $\delta \in \{1,2\}$ with $\delta=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we…

Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these…

In this paper, we classify three-dimensional complex Abelian varieties isogenous to a product $A_1 \times A_2$, where one of the factors admits real multiplication by a real quadratic order $\mathcal{O}_D$ of discriminant $D$. We show that…

It is well known that for any prime $p\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a…

Deep work by Shintani in the 1970's describes Hecke $L$-functions associated to narrow ray class group characters of totally real fields $F$ in terms of what are now known as Shintani zeta functions. However, for $[F:\mathbb{Q}] = n \geq…

We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their…

Inspired by the work of G. Harder (\cite{HaICM}, \cite{HaLNM}, \cite{HaMM}) we construct via the motive of a Hilbert modular surface an extension of a Tate motive by a Dirichlet motive. We compute the realisation classes and indicate how…

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_\mathfrak{p}$ acting on $(\mathrm{mod}\, p^m)$ Katz Hilbert modular classes which agrees with the…

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.

We consider the integers $\alpha$ of the quadratic field $ \mathbb{Q} (\sqrt{d}$ $)$ where $d\in \Z$ is square-free and $d\equiv 1,2,3 \pmod 4$. Let $p$ be an odd prime. Using the embedding into $ \text{GL}(2,\mathbb{Z})$ we obtain bounds…

In this paper, we determine the 2-rank of the class group of certain classes of real cyclic quartic number fields. Precisely, we consider the case in which the quadratic subfield is Q(\sqrt{l}) with l=2 or a prime congruent to 1 mod 8.

Andrews, Dyson, and Hickerson showed that 2 $q$-hypergeometric series, going back to Ramanujan, are related to real quadratic fields, which explains interesting properties of their Fourier coefficients. There is also an interesting relation…

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of…

Let $K$ be an imaginary quadratic field of discriminant less than or equal to -7 and $K_{(N)}$ be its ray class field modulo $N$ for an integer $N$ greater than 1. We prove that singular values of certain Siegel functions generate $K_{(N)}$…

We use Borcherds products and their restrictions to Hirzebruch-Zagier curves to determine generators and relations for the graded rings of Hilbert modular forms for the fields $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$. These seem…

We give a general formula of the bias of root numbers for Hilbert modular newforms of cubic level. Explicit calculation is given when the base field is $\mathbb{Q}, \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5})$ and the level is the cube of…